Problem 66 - Precalculus: Functions and Graphs

Question

Graph the equation on the Interval \([-2,2]\), and describe the behavior of \(y\) as \(x \rightarrow 0^{-}\) and as \(x \rightarrow 0^{+}\) \(y=\frac{1-\cos 3 x}{x}\)

Step-by-Step Solution

Answer

As \( x \rightarrow 0^- \) and \( x \rightarrow 0^+ \), \( y \) approaches 0 smoothly without a discontinuity.

Step 1: Understanding the Function

The function given is \( y = \frac{1 - \cos 3x}{x} \). It's a combination of a trigonometric function and a rational expression, with a potential for an undefined value at \( x = 0 \). We need to analyze this for the interval \([-2, 2]\).

Step 2: Behavior Analysis

To understand the behavior of the function \( y \) as \( x \to 0^− \) and as \( x \to 0^+ \), we look at the limits. The function can be analyzed using the limit definition and L'Hopital's Rule as it has the form \( \frac{0}{0} \) at \( x = 0 \).

Step 3: Applying L'Hopital's Rule

Since the direct substitution of \( x = 0 \) leads to an indeterminate form \( \frac{0}{0} \), we use L'Hopital's Rule. This requires us to differentiate the numerator and the denominator. The derivative of \( 1 - \cos 3x \) is \( 3\sin 3x \), and the derivative of \( x \) is \( 1 \). Thus, the limit relies on \( \lim_{x \to 0} \frac{3\sin 3x}{1} = \lim_{x \to 0} 3\sin 3x = 0 \).

Step 4: Graphing the Function

Plot the graph of \( y = \frac{1 - \cos 3x}{x} \) within the interval \([-2, 2]\), making sure to identify key points including the limits at zero and symmetrical points due to the periodic cosine. Note that the behavior near zero suggests a smooth transition approaching 0 from either side.

Step 5: Conclusion on Function's Behavior

As \( x \rightarrow 0^- \) and \( x \rightarrow 0^+ \), the value of \( y \) approaches 0, meaning the function is continuous and smooth around \( x = 0 \). The graph is perfectly smooth and does not have a discontinuity at zero, based on L'Hopital's adjustment.

Key Concepts

L'Hopital's Rule

L'Hospital's Rule is a powerful tool used when you are tackling indeterminate forms, which typically appear in the form of \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When you see these forms while taking the limit of a function, L'Hopital's Rule can help. The rule states that if you have a limit of the form \( \lim_{x \to c} \frac{f(x)}{g(x)} \) equal to \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), you can differentiate the numerator and the denominator separately, then take the limit again. This often simplifies the problem and allows you to find the limit. For example, to evaluate \( \lim_{x \to 0} \frac{1 - \cos 3x}{x} \), after differentiating the numerator and denominator, you get \( \lim_{x \to 0} \frac{3\sin 3x}{1} \), which equals 0, demonstrating a continuous function around the point \( x = 0 \). This rule is invaluable in calculus, especially when dealing with functions involving trigonometric elements.

Limits and Continuity

Understanding limits and the concept of continuity is crucial, especially when dealing with functions that may initially seem undefined at a certain point, such as \( x = 0 \) in the function \( y = \frac{1 - \cos 3x}{x} \). A limit tells us what value a function approaches as the input approaches some value. Continuity means that as \( x \) approaches a particular point, the function's value approaches the function's actual value at that point (if it exists). In this exercise, by calculating the limit as \( x \) approaches zero from both negative and positive directions, we establish that \( y \) approaches 0 from both sides, signifying a smooth and consistent transition through that point. This is a classic example of how limits confirm the continuity of a function at what appears initially to be a problem point.

Trigonometric Identities

Trigonometric identities are formulas that relate the angles and sides of a triangle. In calculus, they are extremely useful in simplifying and evaluating expressions involving sine, cosine, tangent, and other trigonometric functions. In the function \( y = \frac{1 - \cos 3x}{x} \), the identity properties of cosine are used. Specifically, knowing that \( 1 - \cos \theta \) can simplify certain calculations is crucial. For instance, the identity \( \cos 3x = 1 - 2\sin^2 \frac{3x}{2} \) can help in understanding the behavior of trigonometric components when differentiating the function. These identities thus play a key role in calculus by making complex expressions more manageable and providing insight into the function's behavior across its domain.